
in which case it is proved in [1]} that \(d_{[]}\)  is a cone distance.
The weight \((t) = -\log \cos ^2(\min (\tfrac{\pi }{2}, t))\)  might seem more peculiar but it arises from the dynamical formulation of section REF . The Wasserstein-Fisher-Rao metric is actually the length space induced by the Gaussian-Hellinger distance (if the ground metric \(d_\)  is itself geodesic), as proved in [2]}, [3]}.
This weight introduces a cut-off, because \((d_(x,y))=+\infty \)  if \(d_(x,y)>\pi /2\) . There is no transport between points too far from each other. The choice of \(\pi /2\)  is arbitrary, and can be modified by scaling \(t \mapsto (t/\rho )\)  for some cutoff \(\rho >0\) . Note it differs slightly from taking \(\phi = \rho \) .
